## Ohio State University Algebraic Geometry Seminar## Year 2018-2019Time: Tuesdays 3-4pmLocation: MW 154 |
---|

(Tseng): Let S be a nonsingular surface. A version of the crepant resolution conjecture predicts that the descendant Gromov-Witten theory of Hilb^{n}(S), the Hilbert scheme of n points on S, is equivalent to the descendant Gromov-Witten theory of Sym^{n}(S), the n-fold symmetric product of S. In this talk we discuss how this works when S is **C**^{2}. We explicitly identify a symplectic transformation equating the two descendant Gromov-Witten theories. We also establish a relationship between this symplectic transformation and the Fourier-Mukai transformation which identifies the (torus-equivariant) K-groups of Hilb^{n}(**C**^{2}) and Sym^{n}(**C**^{2}). This is based on joint work with R. Pandharipande.

(Webb): When a complex reductive group G with maximal torus T acts on an affine variety S, one can form two (GIT) quotients: W//G and W//T. With the right hypotheses, W//G and W//T are both smooth projective varieties. The relationship between the cohomology rings of these two varieties is well understood. I will discuss the relationship of their quantum cohomology using the quasimap theory of Ciocan-Fontanine and Kim. In particular, I will present the abelian-nonabelian correspondence for quasimap I-functions when W is a vector space and G is connected.

(Cueto): Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement "any smooth surface of degree three in **P**^{3} contains exactly 27 lines'' is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in **TP**^{3}.
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in **P**^{44} via its anticanonical bundle. The combinatorics of the root system of type E_{6} and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.

(Richman): The set of (higher) Weierstrass points on a curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on tropical curves (i.e. finite metric graphs), and explain how their distribution can be described in terms of electrical networks. Knowledge of tropical curves will not be assumed, but knowledge of how to compute resistances (e.g. in series and parallel) will be useful.

(Hast): Given a curve of genus at least 2 over a number field, what can we say about its set of rational points? Faltings' theorem tells us that this set is finite, but many questions remain about how to obtain good bounds on the number of rational points and how to provably list all rational points. We will survey some recent progress and ongoing work on these questions using Kim's non-abelian Chabauty method, which uses the fundamental group to construct *p*-adic analytic functions that vanish on the set of rational points.

In particular, we present a new proof of Faltings' theorem for
superelliptic curves over **Q**, due to joint work with Jordan
Ellenberg. We will also discuss a conditional generalization of this
strategy from **Q**-points to points in any real number field.

(Yang): Given a Fano variety X over a number field, and a height function on X. Manin's conjecture predicts the asymptotic growth of the number of rational points on X of bounded height. When X is an equivariant compactification of a homogeneous variety, the conjecture is related to equidistribution of adelic periods. We will discuss some recent progress in this direction.

(Ullery): The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. We will discuss these measures of irrationality and various methods of calculating and bounding them. We’ll mainly focus on the examples of hypersurfaces and, more generally, complete intersections in projective space.

(Doran): The quest to understand the commonalities and differences between topology and algebraic geometry is central to many mathematical developments. Here we consider a number of seemingly unrelated but well-known unsolved problems (in geometry, number theory, representation theory): for instance, dating back to Newton's time, what is the minimal degree curve passing through n chosen points in the plane with multiplicities? Relatedly, a common, but famously difficult, broad question is to determine when a ``homology" class is represented by a (symplectic, algebraic, etc.) geometric object, that is to say, when is a class effective?

We recast these problems using a canonical construction that crucially exploits an oft-overlooked difference between topological and algebraic structures. The resulting "uncloaked" problems can then all be attacked with the same weapon. At least up to scaling, this is a form of stability analysis using both additive and multiplicative groups. Beautiful combinatorial structures, like scuffed polytopes and modifications of Okounkov bodies arise naturally. The results are suggestive that effectivity may reduce to stability much more broadly still, and that many wall-crossings in geometry may admit interpretation as a change in this form of stability.

If there is time, we may briefly mention links with quantum entanglement and entropies, and question whether physics is sensitive to some of these differences between topology and algebraic geometry.

Some of this work is joint with Frances Kirwan.

(Denham): The maximal likelihood variety of a complex hyperplane arrangement describes the set of critical points of all rational functions with poles and zeros on the arrangement. This variety's bidegree encodes the h-vector of the underlying matroid's broken circuit complex. I will describe work-in-progress with Federico Ardila and June Huh that constructs a tropical version of the maximal likelihood variety. This sheds some new light on the tropical characteristic classes of López de Medrano, Rincón, and Shaw, as well as on the h-vector of the broken circuit complex of an arbitrary matroid.

**Ohio State University Algebraic Geometry Seminar-Year 2017-2018**

**Ohio State University Algebraic Geometry Seminar-Year 2016-2017**

**Ohio State University Algebraic Geometry Seminar-Year 2015-2016**

**Ohio State University Algebraic Geometry Seminar-Year 2014-2015**

**Ohio State University Algebraic Geometry Seminar-Year 2013-2014**

This page is maintained by Angie Cueto and Dave Anderson.